The weak solution to the Navier–Stokes equations in a bounded domain D ⊂ R[superscript 3] with a
smooth boundary is proved to be unique provided that it satisfies an additional requirement.
This solution exists for all t ≥ 0. In a bounded domain D the solution decays exponentially
fast as t → ∞if the force term decays at a suitable rate

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http://www.sciencedirect.com/science/article/pii/S0893965912004065

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Navier-Stokes equations

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Weak solution

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Uniqueness theorem

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Large-time behavior of the weak solution to 3D Navier-Stokes equations